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Introduction to Small-Angle Neutron Scattering and Neutron Reflectometry

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Introduction to Small-Angle Neutron Scattering and Neutron Reflectometry Introduction to Small-Angle Neutron Scattering and Neutron Reflectometry Andrew J Jackson NIST Center for Neutron Research May 2008 Contents 1 Introduction 2 2 Neutron Scattering 2 2.1 Neutron-nucleus interaction . 2 2.2 Scattering Cross Section . 4 2.3 Coherent and Incoherent Cross Sections . 6 3 Small Angle Neutron Scattering 6 3.1 General Two Phase System . 8 4 Analysis of Small Angle Scattering Data 10 4.1 Model Independent Analysis . 10 4.1.1 The Scattering Invariant . 10 4.1.2 Porod Scattering . 11 4.1.3 Guinier Analysis . 11 4.2 Model Dependent Analysis . 12 4.2.1 The Form Factor for Spheres . 13 4.2.2 The Form Factor for Cylinders . 14 4.3 Contrast Variation . 15 4.4 Polydispersity . 15 5 Neutron Reflectometry 16 5.1 Specular Reflection . 17 5.1.1 Classical Optics . 18 5.1.2 Interfacial Roughness . 19 5.1.3 Kinematic (Born) Approximation . 20 6 Analysis of Reflectometry Data 20 7 Recommended Reading 22 7.1 Neutron Scattering . 22 1 7.2 Small Angle Neutron Scattering . 22 7.3 Reflectometry . 22 8 Acknowledgements 22 9 References 22 9.1 Scattering and Optics . 23 9.2 Reflectometry . 23 A Radius of Gyration of Some Homogeneous Bodies 24 1 Introduction The neutron is a spin 1/2 sub-atomic particle with mass equivalent to 1839 electrons (1.674928×10−27 −27 −1 kg), a magnetic moment of -1.9130427 µn (-9.6491783×10 JT ) and a lifetime of 15 minutes (885.9 s). Quantum mechanics tells us that, whilst it is certainly particulate, the neutron also has a wave nature and as such can display the gamut of wave behaviors including reflection, refraction and diffraction. This introduction covers briefly the theory of neutron scattering and that of two techniques that make use of the wave properties of neutrons to probe the structure of materials, namely small angle neutron scattering (diffraction) and neutron reflectometry (reflection and refraction). Since this introduction is exactly that, the reader is encouraged to look to the extensive literature on the subject and a recommended reading list is provided at the end. Much of the material presented here has been taken from those references. 2 Neutron Scattering 2.1 Neutron-nucleus interaction The scattering of neutrons occurs in two ways, either through interaction with the nucleus (nuclear scattering) or through interaction of unpaired electrons (and hence the resultant magnetic moment) with the magnetic moment of the neutron (magnetic scattering). It is the former of these that this introduction will address. Let us consider the elastic scattering of a beam of neutrons from a single nucleus. In this case we treat the nucleus as being rigidly fixed at the origin of coordinates and there is no exchange of energy (Figure (1)). The scattering will depend upon the interaction potential V(r) between the neutron and the nucleus, separated by r. This potential is very short range and falls rapidly to zero at a distance of the order of 10−15 m. This is a much shorter distance than the wavelength 2 y k’ Scattered circular wave: -b ikr r r e k x Nucleus at r=0 Incident plane wave: eikx |q| 1 sinθ = k' 2 |k| q q = 2k sinθ = 4πsinθ 2θ λ k Figure 1: Elastic neutron scattering from a fixed nucleus (after Pynn, 1990) of the neutrons which is of the order of 1A˚ (10−10 m) and as a result the nucleus acts as a point scatterer. We can represent the beam of neutrons by a plane wave with wavefunction ikz Ψi = e (1) where z is the distance from the nucleus in the propagation direction and k = 2π/λ is the wave- number. The scattered wave will then be spherically symmetrical (as a result of the nucleus being a point scatterer) with wavefunction b ikr Ψs = − e (2) r where b is the nuclear scattering length of the nucleus and represents the interaction of the neutron with the nucleus. The minus sign is arbitrary and is used so that a positive value for b indicates a repulsive interaction potential. The scattering length is a complex number, but the imaginary component only becomes important for nuclei that have a high absorption coefficient (such as boron and cadmium) and it can otherwise be treated as a real quantity. The scattering length of nuclei varies randomly across the periodic table. It also varies between isotopes of the same element. A useful example of this is 1H and 2H ( hydrogen and deuterium respectively with the latter often labeled D). Hydrogen has a coherent (see later section) scattering length of −3:74 × 10−5A˚ and deuterium 6:67 × 10−5A.˚ Thus the scattering length of a molecule can 3 be varied by replacing hydrogen with deuterium and potentially be made to match that of some other component in the system. This technique of contrast variation is one of the key advantages of neutron scattering over x-rays and light. As mentioned above, the neutron can also interact with the magnetic moment of an atom. This magnetic interaction has a separate magnetic scattering length that is of the same order of magni- tude, but independent from, the nuclear scattering length. Thus, for example, one can use contrast variation to remove the nuclear component of the scattering and leave only the magnetic. The magnetic interaction is spin-dependent, so it is also possible to extract information about the mag- netization through the use of polarized neutrons. These advanced uses are beyond the scope of this introduction, but more information can be found in the reference material listed at the end. Having treated the case of a single nucleus, if we now consider a three-dimensional assembly of nuclei whilst maintaining the assumption of elastic scattering the resultant scattered wave will then be X bi ikr iq·r Ψs = − e e (3) r i where q = ki − ks and is known as the scattering vector with ki and ks being the wavevectors of the incoming and scattered neutrons respectively. 2.2 Scattering Cross Section The scattering cross section is a measure of how \big" the nucleus appears to the neutron and thus how strongly neutrons will be scattered from it. Scattering direction θ , φ dS r φ dΩ Incident Neutrons k θ z-axis Figure 2: The geometry of a scattering experiment (after Squires) Imagine a neutron scattering experiment where a beam of neutrons of a given energy E is incident 4 on a general collection of atoms (your sample - it could be a crystal, a solution of polymers, a piece of rock, etc) (Figure 2). If we again assume elastic scattering (such that the energy of the neutrons does not change) we can set up a neutron detector to simply count all the neutrons scattered into the solid angle dΩ in the direction θ; φ. The differential cross section is defined by dσ number of neutrons scattered per second into dΩ in direction θ; φ = (4) dΩ ΦdΩ where Φ is the number of incident neutrons per unit area per second, referred to as the incident flux. The name \cross section" suggests that this represents an area and indeed, we can see that the dimensions of flux are [area−1 time−1] and those of the numerator in equation (4) are [time−1] resulting in dimensions of [area] for the cross section. The total scattering cross section is defined by the equation total number of neutrons scattered by second σs = (5) Φ and is related to the differential scattering cross section by Z dσ σs = dΩ (6) dΩ The cross section is the quantity that is actually measured in a scattering experiment and the basic problem is to derive theoretical expressions that describe it for given systems of scatterers. Exper- imentally the cross sections are usually quoted per atom or per molecule and thus the definitions above are then divided by the number of atoms or molecules in the scattering system. We can calculate the cross section dσ=dΩ for scattering from a single fixed nucleus using the expressions given above. Denoting the velocity of the neutrons as v and again treating elastic scattering, the number of scattered neutrons passing through an area dS per second is 2 2 b 2 vdSj sj = vdS = vb dΩ (7) r2 The incident neutron flux is 2 Φ = vj ij = v (8) From equation (4) dσ vb2dΩ = = b2 (9) dΩ ΦdΩ and then integrating over all space (4π steradians) we obtain 2 σtot = 4πb (10) We can perform a similar calculation for the assembly of nuclei whose wavefunction was given in equation (3) above and obtain the differential cross section N 2 dσ 1 X iq·r (q) = bie (11) dΩ N i which we can now see is a function of the scattering vector, q. 5 2.3 Coherent and Incoherent Cross Sections The above discussion applies to the case where there is only one isotope of one element present (specifically an element with zero nuclear spin), however practically all real systems will have a distribution of both elements and isotopes of those elements. The result of this distribution is that the total cross section is, in fact, a sum of two components a coherent part and an incoherent part σtot = σcoh + σincoh (12) The coherent scattering cross section, σcoh, represents scattering that can produce interference and thus provides structural information. Conversely, the incoherent cross section does not contain structural information. The two are related to the mean and variance of the scattering length such that 2 2 2 σcoh = 4π < b > and σincoh = 4π(< b > − < b > ) (13) The total scattering cross section is then 2 σs = 4π < b > (14) We previously learned that the scattering length b is, in fact, a complex number.
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